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May 20th, 2017, 07:49 PM   #1
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Problems on Series

1. The largest Interval $I$ such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}} $ converges whenever $\displaystyle x \in I $ is equal to
A)$ [-1,1]$
B)$ [-1,1)$
C)$ (-1,1]$
D)$(-1,1)$

My answer : Options A or B

2. Let $\displaystyle \sum a_n $ be a convergent series. Let $b_n=a_{n+1} - a_n$ for all $\displaystyle n \in N $. Then

A) $\displaystyle \sum b_n $ should also be convergent and $\displaystyle (b_n) \rightarrow 0 $ as $\displaystyle n \rightarrow \infty. $

B) $\displaystyle \sum b_n $ need not be convergent but $\displaystyle (b_n) \rightarrow 0 $ as $\displaystyle n \rightarrow \infty. $

C) $\displaystyle \sum b_n $ is convergent but $(b_n)$ need not tend to zero as $\displaystyle n \rightarrow \infty. $

D) None of the above statements is true.

My answer : Option A

3. Which of the following series converge ?

A) $\displaystyle \sum_{n=1}^{\infty} (\frac{\log n}{n^{1+2\epsilon}} $
B) $\displaystyle \sum_{n=1}^{\infty} (\frac{(\log n)^2}{n^{1+2\epsilon}} $
C) $\displaystyle \sum_{n=1}^{\infty} (\frac{n^2+1}{n^3+n}) $
D) $\displaystyle \sum_{n=1}^{\infty} (1+\frac{1}{n})^n $

I checked Option C, it is convergent and Option D is not convergent. Kindly check if Option A or B are Convergent ?
Help me to solve series with logarithms.

Kindly check and let me know If I'm wrong.
Thanks

Last edited by skipjack; May 20th, 2017 at 10:43 PM.
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May 20th, 2017, 11:25 PM   #2
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1. (B)

2. (A)

3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
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May 20th, 2017, 11:41 PM   #3
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Quote:
Originally Posted by skipjack View Post
1. (B)

2. (A)

3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
Thanks for the answers!
Sorry to confuse you. Can you please check the below series now ?

A) $\displaystyle \sum_{n=1}^{\infty} (\frac{\log n}{n^{1+2e}}) $
B) $\displaystyle \sum_{n=1}^{\infty} (\frac{(\log n)^2}{n^{1+2e}}) $
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May 20th, 2017, 11:56 PM   #4
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Quote:
Originally Posted by skipjack View Post
1. (B)

2. (A)

3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
Can you suggest some videos or books on series where I can get details of how to check if a series is convergent or divergent and the range of n as well as for the values or a.

It's confusing for me to estimate a series converges or not.

Thanks
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May 21st, 2017, 09:44 AM   #5
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C is precisely $\sum \frac1n$ and thus divergent.
The terms of D tend to $e$ and not $0$, so that is divergent.
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